Problem 38 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the point(s) of inflection of the graph of the function. $$ f(t)=(1-t)(t-4)\left(t^{2}-4\right) $$

Step-by-Step Solution

Answer

The process to find the points of inflection involves calculating the first and second derivative, finding potential inflection points, and verifying these points by testing in the second derivative. The final points of inflection are found by substituting the x-values into the original function to find the corresponding y-values.

Step 1: Compute the first derivative

The first derivative \(f'(t)\) is calculated by applying the product rule and the chain rule: \[ f'(t)=(1-t)(t-4)(2t)+(1-t)((t^2-4)(1)+(t-4)(2t))+(t-4)((t^2-4)(1)+(1-t)(2t))\]

Step 2: Compute the second derivative

The second derivative \(f''(t)\) can now be calculated by taking the derivative of \(f'(t)\).

Step 3: Calculate inflection points

This is done by setting \(f''(t)\) equal to zero, and solving for 't': \[ f''(t) = 0 \]This results in a number of potential points of inflection. Each of these points must be tested in the second derivative to see if it changes signs. If it does, then it is a point of inflection.

Step 4: Finalize the points of inflection

The x-coordinates of the points of inflection are the solutions to the equation from step 3. Plug these x-values into the original equation \(f(t)\) to find their corresponding y-values. These pairs are the points of inflection.

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. Essentially, it is the mathematical way of describing the slope of a function's graph. For any function \( f(t) \), taking the derivative \( f'(t) \) enables us to understand how \( f(t) \) behaves. Every time you deal with a derivative, it tells you how the function's value is changing with respect to changes in the input variable, in this case, \( t \).

The process of finding a derivative is fundamental for solving problems related to rate of change, optimization, and finding approximate solutions to complex equations. Knowing how to compute derivatives is crucial as it lays the foundation for further operations like the product rule, chain rule, and second derivatives.

Applying the Product Rule

The product rule is a technique used to take the derivative of a product of two functions. Imagine you have two functions, \( g(t) \) and \( h(t) \), and their product \( f(t) = g(t) \, h(t) \). When you need the derivative \( f'(t) \), the product rule comes into play:

• First, take the derivative of \( g(t) \) and multiply it by \( h(t) \).
• Next, take the derivative of \( h(t) \) and multiply it by \( g(t) \).

Thus, the derivative \( f'(t) \) will be \( g'(t) \, h(t) + g(t) \, h'(t) \).

In the given exercise, the function \( f(t) \) is a product of several expressions, and the product rule allows the derivative to account for all interactions between them. This application is essential when functions are intertwined as products.

Utilizing the Chain Rule

Whenever a function is composed of another function, the chain rule is your go-to method for finding its derivative. Say you have a function \( h(t) = g(f(t)) \). To differentiate \( h(t) \) with respect to \( t \), you use the chain rule, which requires taking the derivative of \( g \) with respect to \( f \), then multiplying it by the derivative of \( f \) with respect to \( t \). The chain rule formula is:

• \( h'(t) = g'(f(t)) \, f'(t) \)

In the original exercise, parts of the function \( f(t) \) are nested within other parts requiring the application of the chain rule to differentiate correctly. This is crucial for managing complex expressions efficiently and accurately. Without the chain rule, derivatives of such nested functions would be cumbersome and difficult to handle.

Exploring the Second Derivative

The second derivative, denoted as \( f''(t) \), gives us insight into the concavity of a function. While the first derivative \( f'(t) \) tells us how \( f(t) \) is changing, \( f''(t) \) indicates how \( f'(t) \) itself changes with \( t \). Essentially, the second derivative reflects the curvature of the original function.

Finding points of inflection, where the function changes concavity, is made possible by the second derivative. You set \( f''(t) = 0 \) to find potential inflection points as these are places where the curvature might change sign. Once found, substituting these points back into the expression of \( f''(t) \) can confirm a sign change, thus identifying true points of inflection. The second derivative is a powerful tool for understanding the shape and behavior of functions beyond mere rates of change.